Mathematical_Recreations-Kraitchik-2e

146 Mathematical Recreations
Another transformation which does not affect the magic
property may be described as the interchange of diagonally
opposite quarters of the square. The description is exact for
squares of even order (Figure 24), but must be modified
slightly for squares of odd order in the manner shown in
Figure 25.
A magic square remains magic when its numbers are subtracted
from any fixed number. In particular, if the square
is normal and of order n, the transformed square will be normal
if every element is subtracted from n 2 + 1. In a normal
magic square of order n, either member of any pair of numbers
whose sum is n 2 + 1 is called a complement of the other.
The square resulting from this last transformation may be
called the complement of the original square.
3. MAGIC SQUARES OF THE THIRD ORDER
To date it has not been possible to give completely general
methods of construction or complete enumeration of
abc
d e f
g h i
FIGURE 26.
magic squares of all orders. The theory of the squares of the
third order is simple and complete, so it seeIllS worth while
to sketch it here.
The magic constant is 15.
We can show at once that the central number must be 5.
Denote the elements of the square as in Figure 26. If we add
together the two orthogonals and two diagonals containing
e, we find
3e + (a + b + c + d + e + f + g + h + i) = 3e + 45 = 60,
whence e = 5. This has the further consequence that a +i =